Stability Analysis of Distributed Estimators for Large-Scale Interconnected Systems: Time-Varying and Time-Invariant Cases

TL;DR

This paper analyzes the stability of distributed estimators in large-scale interconnected systems, providing conditions for stability and convergence in both time-varying and time-invariant cases, supported by theoretical proofs and simulations.

Contribution

It introduces a novel stability analysis framework for distributed estimators using a decoupling approach based on the system's structure and Riccati equations.

Findings

Subsystem stability is independent of the global system under certain decoupling variables.

The distributed estimator is stable if the Riccati equation remains bounded.

The estimator converges to a unique steady state in time-invariant systems.

Abstract

This paper studies a distributed estimation problem for time-varying/time-invariant large-scale interconnected systems (LISs). A fully distributed estimator is presented by recursively solving a distributed modified Riccati equation (DMRE) with decoupling variables. By partitioning the LIS based on the transition matrix's block structure, it turns out that the stability of the subsystem is independent of the global LIS if the decoupling variable is selected as the number of out-neighbors. Additionally, it is revealed that any LIS can be equivalently represented by a Markov system. Based on this insight, we show that the stability decoupling above can also be achieved if the decoupling variable equals the number of in-neighbors. Then, the distributed estimator is proved to be stable if the DMRE remains uniformly bounded. When the LIS is considered time-invariant, and by analyzing the spectral radius of a linear operator, it is proved that the DMRE is uniformly bounded if and only if a linear matrix inequality is feasible. Based on the boundedness result, we also show that the distributed estimator converges to a unique steady state for any initial condition. Finally, simulations verify the effectiveness of the proposed methods.